@ Applied General Topology © Universidad Polité ni a de Valen iaVolume 12, no. 2, 2011pp. 221-225 Density of κ-Box-Produ ts and the existen e ofgeneralized independent familiesStefan Ottmar ElserAbstra tIn this paper we will prove a slight generalisation of the Hewitt-Mar zewski-Pondi zery theorem (theorem 2.3 below) on erning thedensity of κ-box-produ ts. With this result we will prove the existen eof generalized independent families of big ardinality ( orollary 2.5 be-low) whi h were introdu ed by Wanjun Hu.2010 MSC: 54A25, 54B10; Se ondary: 03E05.Keywords: κ-box-produ t, generalized independent family.1. Introdu tionLet d(X) denote the density and w(X) the weight of the topologi al spa e X.De�nition 1.1. Let µ,κ be two ardinals with ℵ0 ≤ κ ≤ µ and {Xi}i∈µ be afamily of topologi al spa es. � κ i∈µXi denotes the κ-box-produ t whi h is indu ed on the full artesian produ t ∏ i∈µ Xi by the anoni al base B = { ⋂ i∈I pr −1 i (Ui);I ∈ P<κ(µ) and Ui is open in Xi}where P<κ(µ) := {I ⊆ µ; |I| < κ}.For κ = ℵ0 the κ-box-produ t is the usual Ty hono�-produ t [8℄ and for κ+ = µ the κ-box-produ t is the full box-produ t mentioned by Kelley [5℄ andBourbaki [1℄. 222 S. O. ElserIn this paper we will dis uss the density of κ-box-produ ts and the onne -tion with in�nite ombinatori s. The lassi al Hewitt-Mar zewski-Pondi zerytheorem states: d ( � ℵ0 i∈2µXi ) ≤ µ for all spa es Xi with d(Xi) ≤ µThis has been proven for separable spa es by E. Mar zewski [6℄ in 1941. In1944 E. S. Pondi zery [7℄ proved a slighty weaker version for Hausdor� spa esand in 1947 E. Hewitt [3℄ proved the general version as stated above.In theorem 2.4 we will prove: d(�κi∈2µXi) ≤ µ <κ for all spa es Xi with d(Xi) ≤ µ2. Density of κ-Box-Produ tsIn this se tion we will prove a generalisation of Theorem 1 in [2℄. To do sowe start with the following de�nition and proposition:De�nition 2.1. Let κ,µ be two in�nite ardinals with µ ≥ κ, {Xi}i∈I a familyof topologi al spa es and for all i ∈ I let Bi be a base of the topology on Xi. W ⊆ ∏ i∈I Xi is alled a µ- ube if for every i ∈ I there exists Wi ⊆ Bi with W = ∏ i∈I ( ⋂ Wi).Proposition 2.2. Let X be a set, µ ≥ κ two in�nite ardinals, {Xi}i∈I afamily of topologi al spa es, {fi : X → Xi}i∈I a family of fun tions and let Wbe a subset of ∏i∈I Xi whi h is a union of µ- ubes.For every ardinal λ < κ and every tuple 〈{xi}i∈λ ; {Ji}i∈λ〉 of families {xi}i∈λ ⊆ X and {Ji}i∈λ ⊆ P(I), where all Ji are pairwise disjun t and not empty, thereexists a subset Q ⊆ W of ardinality less or equal to µ<κ so that for all families {ji;ji ∈ Ji}i∈λ the following holds: ( W ∩ ⋂ i∈λ pr −1 ji (fji (xji)) 6= ∅ ) ⇒ ( Q ∩ ⋂ i∈λ pr −1 ji (fji (xji)) 6= ∅ ) .Proof. For every tuple 〈{xi}i∈λ ; {Ji}i∈λ〉 with |{i ∈ λ; |Ji| > 1}| = 0 the laimis pretty obvious.So we assume that the proposition is valid for ardinals less than ν and let 〈 {xi}i∈λ ; {Ji}i∈λ 〉 be a tuple with |{i ∈ λ; |Ji| > 1}| = ν.Without loss of generality we may assume that |Ji| > 1 for all i ∈ ν and |Ji| = 1for all other i ≥ ν and that there exists at least one family {ji;ji ∈ Ji}i∈λ with W ∩ ⋂ i∈λ pr −1 ji (fji (xji)) 6= ∅.Let p ∈ W be an point so that prji(p) ∈ fji(xi) for all ν ≤ i ∈ λ.Then there exists an J ∈ P≤µ(I) with { q ∈ ∏ i∈I Xi; ∀j ∈ J : prj(q) = prj(p) } ⊆ W.We hoose for all i ∈ ν and ji ∈ (Ji − J) a point qji ∈ fji(xi) and we de�ne apoint q ∈ W as follows: Density of κ-Box-Produ ts 223 pri(q) := { pri(p) , if i ∈ (I − ⋃l∈ν(Jl − J)) qjl , if i = jl and jl ∈ (Jl − J)By the de�nition of q we have q ∈ (W ∩ ⋂ i∈λ pr −1 ji (fji(xi)) ) for every family {ji;ji ∈ Ji}i∈λ su h that for all i ∈ ν: ji ∈ (Ji − J).Now we have to onsider families {ji;ji ∈ Ji}i∈λ with ji ∈ (Ji ∩ J) for atleast one i ∈ λ.We de�ne Σ := { {J∗i }i∈ν ; |{i ∈ κ;J ∗ i = Ji}| < ν ∧ (J ∗ i 6= Ji ⇒ J ∗ i ∈ P1(Ji ∩ J)) } . ⇒ |Σ| ≤ µν ≤ µλ ≤ µ<κFor all σ = {J∗i }i∈ν ∈ Σ we de�ne a family {Jσi }i∈λ as follows: Jσi := { J∗i , if i ∈ ν Ji , if i ≥ νFor all these {Jσi }i∈λ the proposition already holds, so we an hoose a set Qσ ⊆ W with |Qσ| ≤ µ<κ and for all families {ji;ji ∈ Jσi }i∈λ the followingholds: ( W ∩ ⋂ i∈λ pr −1 ji (fji (xji)) 6= ∅ ) ⇒ ( Qσ ∩ ⋂ i∈λ pr −1 ji (fji (xji)) 6= ∅ ) .Let σ = {ji;ji ∈ Ji}i∈ν be a family with W ∩ ⋂i∈λ pr−1ji (fji (xji)) 6= ∅.Then σ ∈ Σ and Qσ ∩ ⋂i∈λ pr−1ji (fji (xji)) 6= ∅.We de�ne Q := {q} ∪ ⋃ σ∈Σ Qσand be ause |Q| ≤ µ<κ this is the set we were looking for. �Theorem 2.3. Let κ and µ be two in�nite ardinals with µ ≥ κ and let �κi∈IXibe a κ-box-produ t with |I| ≤ 2µ and w(Xi) ≤ µ for all i ∈ I.Then d(W) ≤ µ<κ holds for every subset W ⊆ ∏ i∈I Xi whi h is a union of µ- ubes.Proof. Let |I| = 2µ, so we may assume that I = 2µ.Let B∗ be a base of the κ-box-produ t �κi∈µD of the dis rete spa e D = {0; 1}with |B∗| = µ<κ.For all i ∈ 2µ let Bi be a base of the topology on Xi with |Bi| = µ, X be aset with |X| = µ, {fi;fi : X → Bi}i∈2µ be a family of surje tive fun tions and ψ : 2µ → ∏ i∈µ D be a bije tion. We de�ne Σ := { 〈 {xi}i∈λ ; {Ji}i∈λ 〉 ;λ < κ ∧ ∀i,j ∈ λ : xi ∈ X ∧ ∅ 6= Ji ⊆ 2 µ ∧ ψ(Ji) ∈ B ∗ ∧ (i 6= j ⇒ Ji ∩ Jj = ∅)} 224 S. O. Elserand hoose for every σ ∈ Σ a set Qσ ⊆ W with all the properties as statedin proposition 2.2. We de�ne Q := ⋃ σ∈Σ Qσ. Be ause of |B∗| = µ we have |Σ| ≤ µ<κ and therefore |Q| ≤ µ<κ. We will now show that Q is dense in W .Let O be a nonempty open set in W and U an element of the anoni- al base B of �κi∈2µXi with ∅ 6= U ∩ W ⊆ O. Then there exists a set {ji;i ∈ λ} ∈ P<κ(2 µ) and a family {Ui;Ui ∈ Bi}i∈λ with U = ⋂i∈λ pr−1ji (Ui).We an hoose for all i ∈ λ pairwise disjun t open sets B∗i ∈ B∗ with ψ(ji) ∈ B∗iand xi ∈ X with fji(xi) = Ui.Obviously σ := 〈{xi}i∈λ ; {Ji}i∈λ〉 is an element of Σ and we have the ondi-tion ∅ 6= W ∩ ⋂ i∈λ pr −1 ji (fji(xi)), thus Qσ ∩ U 6= ∅ ⇒ Q ∩ O ⊇ Oσ ∩ W ∩ U = Oσ ∩ U 6= ∅Therefore Q is dense in W and we have d(W) ≤ |Q| ≤ µ<κ. �The following is a slight generalisation of the Hewitt-Mar zewski- Pondi zerytheorem:Theorem 2.4. Let κ and λ be two in�nite ardinals with µ ≥ κ and let �κi∈IXia κ-box-produ t with |I| ≤ 2µ and d(Xi) ≤ µ for all i ∈ I.Then d(�κi∈IXi) ≤ µ<κ.Proof. Obviously there is a set D whi h is dense in �κi∈IXi and |pri(D)| ≤ µfor all i ∈ I.Let �κi∈IWi be the κ-box-produ t of dis rete spa es Wi with |Wi| = µ andlet f : ∏i∈I Wi → D be a ontinuous and surje tive fun tion.Be ause ∏ i∈I Wi itself is an union of µ- ubes and due to theorem 2.3 there isa dense subset Q of W with |Q| ≤ µ<κ.Let O be a nonempty open set in �κi∈IXi. Then D ∩ O 6= ∅ and f−1(D ∩ O)is open in �κi∈IWi.So Q ∩ f−1(D ∩ O) 6= ∅ and ∅ 6= f (Q ∩ f−1(D ∩ O)) ⊆ f(Q) ∩ O.Therefore f(Q) is dense in �κi∈IXi and d(�κi∈IXi) ≤ µ<κ. �Following Wanjun Hu we de�ne:De�nition 2.5. Let S be an in�nite set, κ, λ and θ be three ardinals with κ ≥ ℵ0 and λ ≥ 2. A family I = {Iα}α∈τ of partitions Iα = {Iβα;β ∈ λ} of Sis alled a (κ,θ,λ)-generalized independent family, if following holds: ∀J ∈ P<κ(τ)∀f : J → λ : ∣ ∣ ∣ { ⋂ I f(α) α ;α ∈ J } ∣ ∣ ∣ ≥ θWe an now apply 2.4 on this theorem and we re eive the following:Corollary 2.6. Let κ and λ be two in�nite ardinals with µ ≥ κ.On every set with at least µ<κ elements exists a (κ,1,µ)-generalized indepen-dent family of ardinality 2µ.Proof. Let S be a set of ardinality µ<κ.For every family {Xi}i∈µ of topologi al spa es with d(Xi) ≤ λ the following Density of κ-Box-Produ ts 225holds with theorem 2.4: d ( � κ i∈µXi ) ≤ |S|Wanjun Hu proved in theorem 3.2 in [4℄ that this is equivalent to the existen eof a (κ,1,µ)-generalized independent family of ardinality 2µ on S. �A knowledgements. I am very grateful to Prof. Dr. Ulri h Felgner for hissupport and helpful advi e. Referen es[1℄ N. Bourbaki, Livre III: Topologie générale. Chapitre 1: Stru tures topologiques.Chapitre. 2: Stru tures uniformes.(2ième edition), (Hermann & Cie., Paris, 1951) p.72.[2℄ R. Engelking, Cartesian produ ts and dyadi spa es, Fund. Math. 57 (1965), 287�304.[3℄ E. Hewitt, A remark on density hara ters, Bull. Amer. Math. So . 52 (1946), 641�643.[4℄ W. Hu, Generalized independent families and dense sets of Box-Produ t spa es, Appl.Gen. Topol. 7, no. 2 (2006), 203�209.[5℄ J. L. Kelley, General Topology, New York 1955, p. 107.[6℄ E. Mar zewski, Séparabilité et multipli ation artésienne des espa es topologiques, Fund.Math. 34 (1937), 127�143.[7℄ E. S. Pondi zery, Power problems in abstra t spa es, Duke Math. Journ. 11 (1944),835�837.[8℄ A. Ty hono�, Über die topologis he Erweiterung von Räumen, Math. Ann. 102 (1930),544�561. (Re eived De ember 2008 � A epted O tober 2009)Stefan Ottmar Elser (stefan.elser�web.de)Mathematis hes Institut, Eberhard Karls Universität Tübingen, Auf der Mor-genstelle 10, 72076 Tübingen, Germany Density of -Box-Products and the existence of[5pt] generalized independent families. By S. O. Elser